Gauge fixing

R_ξ gauges[edit]

The R_ξ gauges are a generalization of the Lorenz gauge applicable to theories expressed in terms of an action principle with Lagrangian density ${\displaystyle {\mathcal {L}}}$. Instead of fixing the gauge by constraining the gauge field a priori, via an auxiliary equation, one adds a gauge breaking term to the "physical" (gauge invariant) Lagrangian $${\displaystyle \delta {\mathcal {L}}=-{\frac {\left(\partial _{\mu }A^{\mu }\right)^{2}}{2\xi }}}$$


The choice of the parameter ξ determines the choice of gauge. The R_ξ Landau gauge is classically equivalent to Lorenz gauge: it is obtained in the limit ξ → 0 but postpones taking that limit until after the theory has been quantized. It improves the rigor of certain existence and equivalence proofs. Most quantum field theory computations are simplest in the Feynman–'t Hooft gauge, in which ξ = 1; a few are more tractable in other R_ξ gauges, such as the Yennie gauge ξ = 3.


An equivalent formulation of R_ξ gauge uses an auxiliary field, a scalar field B with no independent dynamics: $${\displaystyle \delta {\mathcal {L}}=B\,\partial _{\mu }A^{\mu }+{\frac {\xi }{2}}B^{2}}$$


The auxiliary field, sometimes called a Nakanishi–Lautrup field, can be eliminated by "completing the square" to obtain the previous form. From a mathematical perspective the auxiliary field is a variety of Goldstone boson, and its use has advantages when identifying the asymptotic states of the theory, and especially when generalizing beyond QED.


Historically, the use of R_ξ gauges was a significant technical advance in extending quantum electrodynamics computations beyond one-loop order. In addition to retaining manifest Lorentz invariance, the R_ξ prescription breaks the symmetry under local gauge transformations while preserving the ratio of functional measures of any two physically distinct gauge configurations. This permits a change of variables in which infinitesimal perturbations along "physical" directions in configuration space are entirely uncoupled from those along "unphysical" directions, allowing the latter to be absorbed into the physically meaningless normalization of the functional integral. When ξ is finite, each physical configuration (orbit of the group of gauge transformations) is represented not by a single solution of a constraint equation but by a Gaussian distribution centered on the extremum of the gauge breaking term. In terms of the Feynman rules of the gauge-fixed theory, this appears as a contribution to the photon propagator for internal lines from virtual photons of unphysical polarization.


The photon propagator, which is the multiplicative factor corresponding to an internal photon in the Feynman diagram expansion of a QED calculation, contains a factor g_μν corresponding to the Minkowski metric. An expansion of this factor as a sum over photon polarizations involves terms containing all four possible polarizations. Transversely polarized radiation can be expressed mathematically as a sum over either a linearly or circularly polarized basis. Similarly, one can combine the longitudinal and time-like gauge polarizations to obtain "forward" and "backward" polarizations; these are a form of light-cone coordinates in which the metric is off-diagonal. An expansion of the g_μν factor in terms of circularly polarized (spin ±1) and light-cone coordinates is called a spin sum. Spin sums can be very helpful both in simplifying expressions and in obtaining a physical understanding of the experimental effects associated with different terms in a theoretical calculation.


Richard Feynman used arguments along approximately these lines largely to justify calculation procedures that produced consistent, finite, high precision results for important observable parameters such as the anomalous magnetic moment of the electron. Although his arguments sometimes lacked mathematical rigor even by physicists' standards and glossed over details such as the derivation of Ward–Takahashi identities of the quantum theory, his calculations worked, and Freeman Dyson soon demonstrated that his method was substantially equivalent to those of Julian Schwinger and Sin-Itiro Tomonaga, with whom Feynman shared the 1965 Nobel Prize in Physics.


Forward and backward polarized radiation can be omitted in the asymptotic states of a quantum field theory (see Ward–Takahashi identity). For this reason, and because their appearance in spin sums can be seen as a mere mathematical device in QED (much like the electromagnetic four-potential in classical electrodynamics), they are often spoken of as "unphysical". But unlike the constraint-based gauge fixing procedures above, the R_ξ gauge generalizes well to non-abelian gauge groups such as the SU(3) of QCD. The couplings between physical and unphysical perturbation axes do not entirely disappear under the corresponding change of variables; to obtain correct results, one must account for the non-trivial Jacobian of the embedding of gauge freedom axes within the space of detailed configurations. This leads to the explicit appearance of forward and backward polarized gauge bosons in Feynman diagrams, along with Faddeev–Popov ghosts, which are even more "unphysical" in that they violate the spin–statistics theorem. The relationship between these entities, and the reasons why they do not appear as particles in the quantum mechanical sense, becomes more evident in the BRST formalism of quantization.